Civil Engineering Reference
In-Depth Information
There are still other types of time domain and ensemble statistics that are of great
importance in wind engineering and that have not yet been mentioned. These comprise
the properties of threshold crossing, the distributions of peaks and extreme values, and
finally, the auto and cross spectral densities, which are frequency domain properties of
the process, i.e. they are frequency domain counterparts to the concepts of variance and
covariance. These are dealt with below.
2.3 Threshold crossing and peaks
()
In Fig. 2.8 is illustrated a time series realisation
x t
of a Gaussian stationary and
homogeneous process (for simplicity with zero mean value), taken over a period
T
. First
we seek to develop an estimate of the average frequency
()
f
a
between the events that
()
x t
is crossing the threshold
a
in its upward direction.
Let a single upward crossing take place in a time interval
t
Δ
that is small enough to
justify the approximation
(
)
()
()
x t
t
xt
xt
t
+Δ
≅
+
⋅Δ
(2.28)
Δ
The probability of an up crossing event during
is then given by
()
()
()
( )
P
xt
a
and
xt
xt
t a
>
f a
t
⎡
≤
+
⋅ Δ
⎤
=
⋅ Δ
(2.29)
⎣
⎦
x
from which it follows that
a
∞
⎡
⎤
1
()
(
)
f
a
lim
∫∫
p
x x dx dx
,
=
(2.30)
⎢
⎥
x
xx
t
Δ
t
0
Δ→
⎢
⎥
⎣
⎦
0
ax t
−⋅Δ
(
)
()
()
p
xx
,
is the probability density function for the joint events
x t
and
x t
.
where
xx
t
0
As
Δ→
the following approximation applies
a
(
)
(
)
∫
p
xx dx
,
x
t p
ax
,
≅⋅ Δ⋅
(2.31)
xx
xx
ax t
−⋅Δ
